crystal oscillators

1. Crystal
OSCILLATOR

2. Introduction:
• Tuned circuit oscillators (LC type) typically Hartley,
Colpitts oscillators produce pure sinusoidal signal
owing to their high Q tank circuit.
• However, practically there is an upper limit in
improving Q of tank beyond a certain value as
there is a finite resistance associated with
inductor and capacitor of tank circuit.
• Thus high frequency oscillators suffer from poor
frequency stability due to limitation in achieving
high Q of tank circuit.
• To obtain a very high level of oscillator stability
a Quartz Crystal is generally used as the
frequency determining device which leads to
another types of oscillator circuit known generally
as a Quartz Crystal Oscillator, (XO).
Crystal Oscillator
Quartz Crystal

3. Description:
• When a voltage source is applied to a small thin piece of
Quartz Crystal, it begins to change shape producing a
characteristic known as the Piezo-electric effect.
• This Piezo-electric Effect is the property of a crystal by
which an electrical charge produces a mechanical force
by changing the shape of the crystal and vice versa, a
mechanical force applied to the crystal produces an
electrical charge.
• Thus piezo-electric crystal, such as quartz exhibits
electro-mechanical resonance characteristics.
• The resonance frequency of the piezo crystal is mainly
decided by the dimensions of it. Since it is a mechanical
resonance, it is characterized by a very high Quality factor
of the order of 20000 or more that can lead to obtaining
frequencies of the order of few Mega Hertz to Few tens
of Mega Hertz.
Crystal Oscillator

4. Description
• The quartz crystal used in a Quartz Crystal Oscillator is a
very small, thin piece or wafer of cut quartz with the two
parallel surfaces metallised to make the required
electrical connections.
• The physical size and thickness of a piece of quartz crystal
is tightly controlled since it affects the final or
fundamental frequency of oscillations. The fundamental
frequency is generally called the crystals “characteristic
frequency”.
• This characteristic frequency is inversely proportional to
its physical thickness between the two metallised
surfaces.
• Once cut and shaped, the crystal can not be used at any
other frequency. In other words, its size and shape
determines its fundamental oscillation frequency.
• The electrical symbol and the electrical equivalent circuit
of the piezo-crystal is shown in the figure.
Crystal Oscillator
Quartz Crystal

5. Analysis
• A mechanically vibrating crystal can be
represented by an equivalent electrical circuit
consisting of low resistance R, a large inductance
L and small capacitance C in series, which is in
parallel with Cp as shown in the figure.
• Cp is the parasitic capacitance between the
metallic plates in which the crystal is placed.
• If we ignore the resistance shown in equivalent
circuit, the impedance of the crystal is given as,
𝑍 𝑠 = (𝑋 𝐿+𝑋 𝐶𝑠) 𝑋 𝐶𝑝 --- (1)
⇒ 𝑍 𝑠 = 𝑗𝜔𝐿 +
1
𝑗𝜔𝐶 𝑠
1
𝑗𝜔𝐶 𝑝
--- (2)
• Substituting 𝑠 = 𝑗𝜔 in the above equation and
re-arranging, we get,
𝑍 𝑠 =
1
𝑠𝐶 𝑝
𝑠2+
1
𝐿𝐶 𝑠
𝑠2+
𝐶 𝑠 𝐶 𝑝
𝐶 𝑠+ 𝐶 𝑝
--- (3)
Crystal Oscillator

6. Analysis
• Substituting 𝑠 = 𝑗𝜔 in the above equation and
re-arranging, we get,
𝑍 𝑠 =
1
𝑠𝐶 𝑝
𝑠2+
1
𝐿𝐶 𝑠
𝑠2+
𝐶 𝑠 𝐶 𝑝
𝐶 𝑠+ 𝐶 𝑝
--- (3)
• From Equation 3, one can observe that the
crystal can oscillate or resonate at two different
frequencies.
• The first resonance frequency occurs when
impedance becomes zero, i.e. when numerator
of equation 3 becomes zero.
𝑠2 +
1
𝐿𝐶 𝑠
= 0 --- (4)
⇒ −𝑗𝜔2 +
1
𝐿𝐶 𝑠
= 0 --- (5)
⇒ 𝝎 = 𝝎 𝒔 =
𝟏
𝑳𝑪 𝒔
--- (6)
• This frequency is called series resonance
frequency.
Crystal Oscillator

7. Analysis
• The second resonance frequency occurs when
the impedance becomes infinite, i.e. when
denominator of equation 3 becomes zero.
𝑠2 +
𝐶 𝑠 𝐶 𝑝
𝐶 𝑠+ 𝐶 𝑝
= 0 --- (7)
⇒ −𝑗𝜔2 +
𝐶 𝑠 𝐶 𝑝
𝐶 𝑠+ 𝐶 𝑝
= 0 --- (8)
⇒ 𝝎 = 𝝎 𝒑 =
𝟏
𝑳𝑪 𝒔 𝑪 𝒑
𝑪 𝒔+ 𝑪 𝒑
--- (9)
• This frequency is called parallel resonance
frequency.
• Please note that at this frequency, the
impedance of the crystal becomes infinite and
hence this is also called “anti-resonance
frequency”
Crystal Oscillator

8. Analysis
• The impedance vs frequency of the crystal is as
shown in figure.
• At a particular frequency, the interaction of
between the series capacitor Cs and the
inductor L creates a series resonance circuit
reducing the crystals impedance to a minimum
and equal to Rs. This frequency point is called
the crystal’s series resonant frequency fs (ωs)
and below fs (ωs) the crystal is capacitive.
• As the frequency increases above this series
resonance point, the crystal behaves like an
inductor until the frequency reaches its parallel
resonant frequency ƒp (ωp).
• At this frequency point the interaction between
the series inductor, Ls and parallel capacitor, Cp
creates a parallel tuned LC tank circuit and as
such the impedance across the crystal reaches
its maximum value.
Crystal Oscillator

9. Analysis
• Then we can see that a quartz crystal is a
combination of a series and parallel tuned
resonance circuits, oscillating at two different
frequencies with the very small difference
between the two depending upon the cut of the
crystal.
• Also, since the crystal can operate at either its
series or parallel resonance frequencies, a
crystal oscillator circuit needs to be tuned to
one or the other frequency as you cannot use
both together.
• So depending upon the circuit characteristics, a
quartz crystal can act as either a capacitor, an
inductor, a series resonance circuit or as a
parallel resonance circuit.
Crystal Oscillator

10. Crystal Oscillator
Oscillator with crystal operating in series resonance.
• Oscillator circuit with crystal operating in series
resonance is shown in figure.
• In this mode of operation crystal impedance is the
smallest and the amount of positive feedback is the
largest.
• Voltage feedback signal from collector to base is
maximum when the crystal impedance is minimum
(i.e. in series resonant mode)
• The resulting circuit frequency of oscillations is set
by the series resonant frequency of the crystal.
• Variations in the supply voltage, transistor
parameters, etc. has no effect on the circuit
operating frequency as this has been stabilized by
the crystal.
Crystal

11. Crystal Oscillator
Oscillator with crystal operating in parallel resonance.
• Oscillator circuit with crystal operating in parallel
resonance is illustrated in figure.
• This is modified version of Colpitts oscillator circuit,
with inductor replaced by Crystal.
• The working of this circuit is same as the Colpitts
oscillator circuit.
• The crystal (Y1) in parallel with C1 and C2 acts as
tank circuit that is responsible for oscillations.
• Variations in the supply voltage, transistor
parameters, etc. has no effect on the circuit
operating frequency as this has been stabilized by
the crystal.
• This type of oscillator is normally called as Pierce
Oscillator. Now we will proceed with design
procedure of this oscillator.
Crystal

12. Crystal Oscillator - Design
Example:
Design and simulate Pierce oscillator shown in the figure to oscillate at 10MHz frequency
using transistor BC547B with a supply voltage of 12V.
• Design Inputs:
• Transistor – BC547A
• Frequency of oscillation – 10MHz.
• DC voltage – 12V.
• Design Steps: Same as the design steps followed in Colpitts oscillator
• For given transistor Q1, BC547A,
• Current gain bandwidth from datasheet = 100MHz which is good for designing
10MHz as per our requirement.

13. Crystal Oscillator
• Design Steps:
• For given transistor Q1, BC547A,
• Current gain bandwidth from datasheet = 100MHz which is good for designing
10MHz as per our requirement.
• ℎ 𝑓𝑒 value of Q1 is 150 @25°C at 10mA which is sufficient for our design.
• First step is to fix the operating point of transistor Q1, BC547A towards which the
following assumptions were made
• RFC can be replaced with resistance Rc = 1KΩ
• Emitter resistance RE = 330Ω.
• Quiescent current = 5mA.
• In order to bias the transistor at the exact midpoint of the load line, the
collector current Ic was calculated as Ic = 4.08mA.
• Considering the stability factor of 10, the biasing resistors calculated to
nearest values as R1 = 20KΩ and R2 = 6.8KΩ for a calculated base resistance of
Rb = 4.36KΩ.
• For an assumed value of RE as 330Ω, the Capacitance CE can be arrived as
10nF.

14. Crystal Oscillator
Design Steps:
• Next step is to arrive at the ratio of capacitances
needed
𝑪 𝟐
𝑪 𝟏
< 𝒈 𝒎 𝑹
• The value of R includes the parallel combination of
collector resistance, base bias resistance, ℎ𝑖𝑒 (or rπ)
load resistance
• For Ic of 4.08mA, the value of the base current
required was found as 0.016mA.
• Hence the value of ℎ𝑖𝑒 is calculated as
ℎ𝑖𝑒 =
25
0.016
= 1.56𝐾Ω.

15. Colpitts Oscillator
Design Steps:
• Considering collector resistance of 1KΩ (refer
figure), R can be calculated as
𝑹 = 𝑹 𝒄 || 𝑹 𝒃 || 𝒉𝒊𝒆 || 𝑹 𝑳
(Please note that a bypass capacitor of 0.1uF is needed
to provide ac ground).
• Substituting the above values, R was calculated as
𝟏
𝑹
=
𝟏
𝟏
+
𝟏
𝟒.𝟑𝟔
+
𝟏
𝟏.𝟓𝟔
+
𝟏
𝟏
= 𝟐. 𝟖𝟕
Hence, R = 0.348KΩ.
• The value of transconductance gm is
𝑰 𝒄
𝑽 𝑻
, Where VT is
generally taken as 25mV at room temp.
𝒈 𝒎 =
𝟒. 𝟎𝟖𝒎𝑨
𝟐𝟓𝒎𝑽
= 𝟎. 𝟏𝟔𝟑𝟐
Hence, 𝒈 𝒎 𝑹 = 𝟓𝟔. 𝟖
• Therefore,
𝑪 𝟐
𝑪 𝟏
= 𝒈 𝒎 𝑹 = 𝟓𝟔. 𝟖

16. Crystal Oscillator
• Design Steps:
• The crystal selected for this design is – Quartz crystal
HC49/U. The data sheet of the same is as shown in
figure.
• The ratio of the values of the parasitic to series
capacitance depends upon the package of crystals and
the resonance frequency of the crystal (fs). For crystal
of HC 49/U, this ratio for a 10 MHZ crystal is around
200.
• Hence given the value of the parasitic capacitance (Co)
as 7 pF as per data sheet shown above, the value of the
series capacitance will be
𝐶𝑠 =
7
200
= 0.035𝑝𝐹
• Now we will arrive at the values of C1 and C2.

17. Crystal Oscillator
• Design Steps:
• The C1, C2 of the circuit diagram comes in series with
the crystal as per the explanation given in the Colpitts
oscillator section.
• The effect of these capacitors C1, C2 will shift the
series resonance frequency closer to parallel
resonance frequency. Hence these values should be
chosen much higher than shunt or parasitic
capacitance of Crystal.
• As discussed in Colpitts oscillator design section, let us
choose the ratio of
𝐶2
𝐶1
as 50.
• Assume the value of C1 as 30pF, we arrive C2 as
1500pF.
• The combined values of C1 and C2 in series will still be
closer to (29.42pf ~=) 30pF which is much greater than
the series capacitance 𝐶𝑠 = 0.035𝑝𝐹and hence it is
not expected to shift the series resonance frequency
• Hence the following are the final
values of tank circuit
• Crystal – HC49/U – 10MHZ.
• C1 = 30pF.
• C2 = 1500pF.

18. Crystal Oscillator – Simulation
• For the given example, the following simulation circuit was built in LT-Spice
software.
• The results are as shown below:
• The simulation results shows that the
frequency of oscillations is very near to the
designed frequency.
• Since the output of the oscillator is nearly a
pure sine wave, it is used widely in RF
systems as local oscillators.
• One can change the values of tank circuit for
obtaining different frequency of oscillations
and can simulate the same for verification.

19. Crystal Oscillator
Lab Experiment:
• Let us know proceed with lab experiment for
Colpitts Oscillator to verify the designed and
simulated circuit that was explained in the
pervious sections.
• One can build the circuit shown in the figure
on a breadboard or general purpose PCB for
conducting experiment.
• Components Required for building the circuit
is as given in the table.
S.No Components Qty
1 Bread board 1
2 DC regulated power supply, 12V/1A 1
3 Resistors
330Ω, 0.25W 1
1KΩ, 0.25W 1
6.8KΩ, 0.25W 1
20KΩ, 0.25W 1
4 Capacitors
0.1uF 2
10nF 1
30pF 1
1500pF 1
5 Crystal
HC49/U – 10MHz 1
6 Transistor
BC547B 1
7 Connecting wires
8 Oscilloscope 1
9 Multimeter 1

20. Crystal Oscillator
Experiment Procedure:
• Connect the circuit, as the schematic shown in
the figure, on a bread board.
• Identify Base, Collector, Emitter of transistor
correctly as per the datasheet and connect on
the breadboard.
• Connect all the components on the breadboard
as per the schematic.
• Power up the circuit using 12V DC supply.
• Connect the oscilloscope at the output of the
circuit.
• Switch on the DC power supply.
• Observe the frequency of the oscillator in an
oscilloscope.
Schematic of Pierce Crystal
oscillator

21. Crystal Oscillator
Observation table:
Schematic of Pierce Crystal oscillator
S.No Crystal C1 C2 Freq (Hz) Amplitude (Pk-Pk)
Conclusion:
• Please put your remarks on the observations: